M-ary signaling can be regarded as a waveform coding procedure, and refers specifically to signal processing where the processor accepts k data bits at a time and instructs the modulator to produce one of M=2k waveforms. Binary signaling (M=2) is the special case where k=1. Typically, M-ary refers to non-binary and that convention is continued throughout this disclosure, but the distinction is also made explicit in certain instances. For pulse amplitude modulation (PAM), the signaling order M refers to the number of unique discrete amplitude values over which the pulse is allowed to vary. Instead of transmitting a pulse waveform for each bit (where the rate would be R bits per second), parse the data into k-bit groups and then use M=2k-level pulses for transmission. Each pulse waveform then represents a k-bit symbol moving at the rate of R/k symbols per second, which reduces the required bandwidth as compared to pulse-modulating each bit because symbols are transmitted as opposed to bits, though at a rate slower by a factor of k. As M (and k) increases, the receiver finds it more difficult to distinguish between, for example, octal pulses (M=8) versus binary pulses (M=2). The result in the prior art is that, as k increases (higher order M-ary signaling) with orthogonal signaling, error performance increases or the required signal to noise ratio SNR (technically Eb/N0) is reduced, at the expense of bandwidth. For non-orthogonal signaling, the tradeoff is reversed in that increasing k improves bandwidth at the expense of lower error performance or increased SNR.
Constant phase modulation (CPM, also known as continuous phase modulation due to its smooth phase transitions between symbols) is a signal modulation technique that increases bandwidth efficiency by smoothing the waveforms in the time domain. Bandwidth efficiency is gained by concentrating the signal's energy in a narrower bandwidth, enabling adjacent signals to be packed closer together. Inherent in this smoothness is the fact that symbol transition features are muted, and many symbol synchronization schemes depend on those transition features being definite. To smooth the time domain signal, various CPM techniques generally rely on one or more of the following features: using signal pulses with several orders of continuous derivatives; intentionally injecting some intersymbol interference so that individual pulses occupy more than one signal time interval; and reducing the maximum allowed phase change per symbol interval.
As a cursory description of CPM, a binary single-h CPM waveform can be expressed over the nth symbol interval as
                                          s            ⁡                          (                              t                ,                a                ,                h                            )                                =                      exp            ⁢                          {                              j                ⁢                                                                  ⁢                2                ⁢                π                ⁢                                                                  ⁢                h                ⁢                                                      ∑                                          i                      =                                              -                        ∞                                                              n                                    ⁢                                                                          ⁢                                                            a                      i                                        ⁢                                          q                      ⁡                                              (                                                  t                          -                                                      ⅈ                            ⁢                                                                                                                  ⁢                            T                                                                          )                                                                                                        }                                      ,                                  ⁢                              nT            ≤            t            <                                          (                                  n                  +                  1                                )                            ⁢              T                                ;                                    [        1        ]            where t denotes time, T denotes the symbol duration, ai∈ {±1} are the binary data bits and h is the modulation index. The modulation index h is the ratio of the frequency deviation to the frequency of the modulating wave, when using a sinusoidal wave as in CPM. The phase function, q(t), is the integral of the frequency function, ƒ(t), which is zero outside of the time interval (0,LT) and which is scaled such that
                                          ∫            0            LT                    ⁢                                    f              ⁡                              (                τ                )                                      ⁢                                                  ⁢                          ⅆ              τ                                      =                              q            ⁡                          (              LT              )                                =                                    1              2                        .                                              [        2        ]            
An M-ary single-h CPM waveform is the logical extension of the binary single-h case in which the information symbols are now multi-level: i.e., ai∈ {±1, ±3, . . . , ±(M−1)}.
Finally, an M-ary multi-h CPM waveform can be written as
                                          s            ⁡                          (                              t                ,                a                ,                h                            )                                =                      exp            ⁢                          {                              j                ⁢                                                                  ⁢                2                ⁢                π                ⁢                                                                  ⁢                                                      ∑                                          i                      =                                              -                        ∞                                                              n                                    ⁢                                                                          ⁢                                                            a                      i                                        ⁢                                          h                      i                                        ⁢                                          q                      ⁡                                              (                                                  t                          -                                                      ⅈ                            ⁢                                                                                                                  ⁢                            T                                                                          )                                                                                                        }                                      ,                                  ⁢                  nT          ≤          t          <                                    (                              n                +                1                            )                        ⁢            T                                              [        3        ]            where ai∈ {±1, ±3, . . . , ±(M−1)} and the modulation index, hn assumes it value over the set: {h(1) . . . , h(Nh)}. In one implementation, for example, the modulation index may cycle over the set of permitted values.
In his seminal work entitled “Exact and Approximate Construction of Digital Phase Modulations by Superposition of Amplitude Modulation Pulses (AMP)”, IEEE Transactions on Communications, vol. COM-34, No. 2, February 1986, pp. 150-160, P. A. Laurent has shown that any binary single-h CPM signal can be exactly represented by the superposition of pulse-amplitude modulation (PAM) waveforms.
                                                                        s                ⁡                                  (                                      t                    ,                    a                    ,                    h                                    )                                            =                              exp                ⁢                                  {                                      j                    ⁢                                                                                  ⁢                    2                    ⁢                    π                    ⁢                                                                                  ⁢                    h                    ⁢                                                                  ∑                        i                                            ⁢                                                                                          ⁢                                                                        a                          i                                                ⁢                                                  q                          ⁡                                                      (                                                          t                              -                                                              ⅈ                                ⁢                                                                                                                                  ⁢                                T                                                                                      )                                                                                                                                }                                                                                                                                          =                                                            ∑                                              k                        =                        0                                                                    Q                        -                        1                                                              ⁢                                                                                  ⁢                                                                  ∑                        n                                            ⁢                                                                                          ⁢                                                                        b                                                      k                            ,                            n                                                                          ⁢                                                  c                          k                                                ⁢                                                  (                                                      t                            -                            nT                                                    )                                                                                                                    ,                                                                  ⁢                                  Q                  =                                                            2                                              L                        -                        1                                                              .                                                              ⁢                                                                                                      [        4        ]            
This is termed the Laurent Decomposition, and ai∈ {±1}, and {bk,n} represents the pseudo-data symbols, which are obtained in a nonlinear fashion from the binary data symbols. Laurent lays the theoretical groundwork in the above paper for representing any constant amplitude binary phase modulation as the sum of a finite number of time-limited pulse amplitude-modulated pulses. Hence, Laurent shows that binary single-h CPM, which may be appear to be rather complex in its classical representation (equation [1]), can be replaced by a much simpler notation by using what has become known as the Laurent Decomposition.
The Laurent Decomposition of equation [4] expresses a binary single-h CPM signal as the sum of 2L−1 PAM waveforms (where L denotes the number of symbol intervals over which its frequency function is defined). The Laurent pulses, ck(t), are obtained from the phase response of the CPM signal. An important characteristic of these pulses is that the signal energy is unevenly distributed amongst them and that the pulses are distinctively ordered. Thus, c0(t) is usually the “main pulse”, which carries most of the signal energy (often upwards of 95%), ci(t) contributes much less energy and the last pulse cQ−1(t) contributes the least amount of energy. Thus, in many cases of practical interest, the CPM waveform can be approximated using only the PAM construction of the “main pulse”
                              s          ⁡                      (                          t              ,              a              ,              h                        )                          ≈                              ∑            n                    ⁢                                          ⁢                                    b                              0                ,                n                                      ⁢                                                            c                  0                                ⁡                                  (                                      t                    -                    nT                                    )                                            .                                                          [        5        ]            
Because the pulses of Laurent's approach are defined in order of decreasing energy, equation [5] can be broadened somewhat to sum the energy over the first pulse or over the first few pulses of the decomposition in order to synthesize an “almost binary single-h CPM” signal. Thus, for many cases of practical interest, the binary single-h CPM signal is well approximated as
                                          s            ⁡                          (                              t                ,                a                ,                h                            )                                ≈                                    ∑                              k                =                0                                                              Q                  n                                -                1                                      ⁢                                                  ⁢                                          ∑                n                            ⁢                                                          ⁢                                                b                                      k                    ,                    n                                                  ⁢                                                      c                    k                                    ⁡                                      (                                          t                      -                      nT                                        )                                                                                      ,                                  ⁢                              1            ≤                          Q              0                        ≤            Q                    =                                    2                              L                -                1                                      .                                              [        6        ]            
The Laurent Decomposition is important because it linearizes the binary CPM waveform, which greatly simplifies receiver algorithms for binary CPM by enabling them to use Laurent's linear approximation of the received CPM signal as a single pulse as in equation [5]. Equation [6] enables simplified design of “almost binary CPM” transmission schemes as well as for the simplification of receiver design algorithms by using only a few of the leading Laurent pulses rather than the true CPM waveform itself.
However, with increasing k (and thus increasing M in an M-ary waveform), further extensions of the Laurent Decomposition do not appear to preserve the mathematically elegant PAM signal structure that makes this decomposition so useful for generating approximations of binary single-h CPM waveforms.
Specifically, in a paper entitled “Decomposition of M-ary CPM Signals Into PAM Waveforms”, IEEE Transactions on Information Theory, vol. 41, No. 5, September 1995, pp. 1265-1275, U. Mengali and M. Morelli extend the Laurent Decomposition to include multi-level (single-h) CPM signaling and show that M-ary single-h CPM waveforms have the following PAM decomposition
                                                                        s                ⁡                                  (                                      t                    ,                    a                    ,                    h                                    )                                            =                              exp                ⁢                                  {                                      j2π                    ⁢                                                                                  ⁢                    h                    ⁢                                                                  ∑                        i                                            ⁢                                                                                          ⁢                                                                        a                          i                                                ⁢                                                  q                          ⁡                                                      (                                                          t                              -                                                              ⅈ                                ⁢                                                                                                                                  ⁢                                T                                                                                      )                                                                                                                                }                                                                                                                        =                                                      ∑                                          k                      =                      0                                                              N                      -                      1                                                        ⁢                                                                          ⁢                                                            ∑                      n                                        ⁢                                                                                  ⁢                                                                  b                                                  k                          ,                          n                                                                    ⁢                                                                        g                          k                                                ⁡                                                  (                                                      t                            -                            nT                                                    )                                                                                                                                ,                                                          ⁢                                                N                  =                                                            Q                      P                                        ⁡                                          (                                                                        2                          P                                                -                        1                                            )                                                                      ;                                                                        [        7        ]            where a now denotes the M-ary data symbols ai∈ {±1, ±3, . . . , ± (M−1)}, Q=2L−1, and P is an integer satisfying the conditions2P−1<M≦2P.   [8]
The Mengali and Morelli approach is seen to view an M-ary CPM signal as the product of P binary CPM waveforms, apply the Laurent Decomposition to each individual factor, and then write the final expression as the sum of PAM components. In general, this approach yields 2P−1 PAM component pulses of significant energy. Furthermore, unlike Laurent's solution for the binary case, their approach does not result in a PAM decomposition in which the component pulses are naturally defined in terms of decreasing signal energy.
E. Perrins and M. Rice also extend the Laurent Decomposition in two papers: “Optimal and Reduced Complexity Receivers for M-ary Multi-h CPM”, Wireless Communications and Networking Conference 2004, pp. 1165-1170; and “PAM Decomposition of M-ary multi-h CPM”, believed to be submitted to IEEE Transactions on Communications for future publication. The work of Perrins and Rice generalize the Laurent Decomposition by applying it to M-ary multi-h CPM waveforms. In their approach, Perrins and Rice first derive the PAM decomposition for the binary multi-h case and then extend this result to the general M-ary multi-h case in order to show that
                                                                        s                ⁡                                  (                                      t                    ,                    a                    ,                    h                                    )                                            =                              exp                ⁢                                  {                                      j2π                    ⁢                                                                  ∑                        i                                            ⁢                                                                                          ⁢                                                                        h                                                      i                            _                                                                          ⁢                                                                                                  ⁢                                                  a                          i                                                ⁢                                                  q                          ⁡                                                      (                                                          t                              -                                                              ⅈ                                ⁢                                                                                                                                  ⁢                                T                                                                                      )                                                                                                                                }                                                                                                                        =                                                      ∑                                          k                      =                      0                                                              N                      -                      1                                                        ⁢                                                                          ⁢                                                            ∑                      n                                        ⁢                                                                                  ⁢                                                                  b                                                  k                          ,                          n                                                                    ⁢                                                                        g                                                      k                            ,                                                          n                              _                                                                                                      ⁡                                                  (                                                      t                            -                            nT                                                    )                                                                                                                                ,                                                          ⁢                              N                =                                                                            Q                      P                                        ⁡                                          (                                                                        2                          P                                                -                        1                                            )                                                        .                                                                                        [        9        ]            
The notation i=i mod Nh, where Nh denotes the number of modulation indexes (i.e. h={h(1), h(2), . . . , h(Nh)} and “mod” indicates modulo addition ⊕. An important distinction between equation [9] and the above work of Mengali and Morelli is that {gk,n(t)} is a now set of Nh·N component pulses. Note that the Perrins-Rice derivation results in the generation of Nh·2P−1 “main pulses” that carry the most significant proportion of the total signal energy.
Each of the above extensions of the Laurent Decomposition is seen to not preserve its mathematical simplicity, which makes it so valuable in mirroring or approximating a binary CPM waveform. What is needed in the art is a method and apparatus to linearly decompose an M-ary CPM signal, with exact or reasonable approximation, so that efficient algorithms and hardware may be designed. This need is seen for both single-h and multi-h modulation. The Laurent Decomposition itself is seen as adequate for binary single-h CPM, so the need lies primarily in the area of non-binary single-h and all multi-h CPM decompositions.